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Theorem topbas 13138
Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
topbas (𝐽 ∈ Top → 𝐽 ∈ TopBases)

Proof of Theorem topbas
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopn 13072 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑦𝐽) → (𝑥𝑦) ∈ 𝐽)
213expb 1204 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) → (𝑥𝑦) ∈ 𝐽)
3 simpr 110 . . . . . . 7 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → 𝑧 ∈ (𝑥𝑦))
4 ssid 3173 . . . . . . 7 (𝑥𝑦) ⊆ (𝑥𝑦)
53, 4jctir 313 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
6 eleq2 2239 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
7 sseq1 3176 . . . . . . . 8 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
86, 7anbi12d 473 . . . . . . 7 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
98rspcev 2839 . . . . . 6 (((𝑥𝑦) ∈ 𝐽 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
102, 5, 9syl2an2r 595 . . . . 5 (((𝐽 ∈ Top ∧ (𝑥𝐽𝑦𝐽)) ∧ 𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1110exp31 364 . . . 4 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → (𝑧 ∈ (𝑥𝑦) → ∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
1211ralrimdv 2554 . . 3 (𝐽 ∈ Top → ((𝑥𝐽𝑦𝐽) → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1312ralrimivv 2556 . 2 (𝐽 ∈ Top → ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
14 isbasis2g 13114 . 2 (𝐽 ∈ Top → (𝐽 ∈ TopBases ↔ ∀𝑥𝐽𝑦𝐽𝑧 ∈ (𝑥𝑦)∃𝑤𝐽 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1513, 14mpbird 167 1 (𝐽 ∈ Top → 𝐽 ∈ TopBases)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2146  wral 2453  wrex 2454  cin 3126  wss 3127  Topctop 13066  TopBasesctb 13111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-uni 3806  df-top 13067  df-bases 13112
This theorem is referenced by:  resttop  13241  txtop  13331
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